Gradually Varied Functions for Advanced Computational Methods

Abstract

Gradually varied functions or smooth gradually varied functions were developed for the data reconstruction of randomly arranged data points, usually referred to as scattered points or cloud points in modern information technology. Gradually varied functions have shown advantages when dealing with real world problems. However, the method is still new and not as sophisticated as more classic methods such as the B-spline and finite elements method. The digital-discrete method has another advantage that is to collaborate with these existing methods to make an even better combined approach in applications. We investigate how the gradually varied function can be applied to more advanced computational methods. We first discuss harmonic analysis, B-spline, and finite element methods. Then we give a new consideration for the smooth function definitions in real world problems. This chapter makes more connections to the mathematical aspects of digital functions.